Basic Measurement Issues
Sampling Theory and Reliability – the extent to which a measurement Analog-to-Digital Conversion procedure yields the same results on repeated trials Validity – the extent to which a measurement procedure measures what it is supposed to measure Introduction/Definitions Sensitivity –responsiveness of an instrument to a Analog-to-digital conversion change in the incoming signal
Sampling Rate Precision – the degree of exactness with which a measurement is made, similar meaning as accuracy Frequency Analysis Resolution – The fineness of detail that can be distinguished in a measurement
Definitions Definitions Signal – a fluctuating electric quantity, such as a voltage or current, whose variations represent Digitization – the process of converting an analog coded information signal into digital form Analog signal – represents data by Digital signal – represents data in Analog-to-digital converter (ADC) – a device that a continuously variable quantity a discrete, numerical form digitizes an analog signal
Sampling – an analog signal is said to be sampled to produce the digital signal
Quantization – to subdivide into small but measurable increments: digital signals are limited by the number of binary digits (“bits”) that can be used to represent them
Analog-to-digital conversion Analog-to-digital conversion process Parameters of the ADC process:
1010 Input range – establishes max voltage range to 1110 0010 0011 be detected 1010 0110 – bipolar: r1V, r5V, r10V – unipolar: 0-1V, 0-5V, 0-10V – usually selectable in hardware or software Input configuration – single-ended: one input relative to ground analog digital computer – differential: difference between 2 inputs is sampled signal ADC signal memory System gain – degree to which signal is amplified before sampling
1 Analog-to-digital conversion ADC – Amplitude Resolution Parameters of the ADC process: A 4-bit A/D set up (24 = 16 discreet levels)
+5 V 1111 binary Amplitude resolution – how precisely amplitude 1110 values of signal can be quantified 1101 – determined by “bit value” of ADC 1100 – common: 12-bit (212 = 4096 discreet levels) 1011 1010 16 16-bit (2 = 65,536 discreet levels) 1001 1000 0 V Temporal resolution – how frequently the ADC 0111 takes samples of the analog signal 0110 – determined by “sampling rate” 0101 – values from 30 Hz to 2000 Hz are common 0100 0011 Together, amplitude and temporal resolution greatly 0010 0001 affect the quality of the sampled signal -5 V 0000
ADC – Amplitude Resolution Analog-to-digital conversion A 4-bit A/D set up – close-up view How precisely does a 12-bit A/D board detect amplitude changes in a signal? 4.375 V 1101 12 bit ADC = 212 = 4096 discreet levels, distributed over the whole input range 3.750 V 1100 For a r10 V input, you have a range of 20 V 3.125 V 1011 (20 V) / (4096 ADC units) = 0.0049 V per ADC unit
2.500 V 1010 Is 4.9mV per ADC unit good or bad resolution? Force Plate: if +10 V = 2500 N (for a given FP gain 1.875 V 1001 setting), then 1 ADC unit = 1.22 N In other words, you will be unable to detect changes in force that are less than 1.22 N, this is probably OK
ADC – Temporal Resolution ADC – Temporal Resolution Sampling Rate: how often is the signal sampled? What is an adequate sampling rate? Sampling period: time between Governed by Shannon’s sampling theorem: adjacent samples “the signal must be sampled at a frequency at least e.g. W = t – t 2 1 twice as high as the highest frequency present in the signal itself” Sampling rate: frequency with This minimally acceptable sampling frequency is which samples known as the Nyquist frequency are taken
e.g. f = 1 / W But what exactly does signal frequency content mean?
t1 t2 t3 …… tn
2 Fourier (Harmonic) Analysis The sine function
• The frequency content of a signal can be determined A – the mean (D.C.) offset from zero by performing a Fourier analysis y(t) A B sin f t B – the amplitude of the signal f– the frequency of the signal • Fourier discovered that any signal, no matter how complex, can be represented by an infinite sum of A = 0 weighted sine and cosine waves B = 1
f f = 1 y(t) A ¦ >Bn sin fn t Cn cos fn t @ n 1 • In practice, only a finite number of harmonics are need to adequately represent a signal
• A related technique, the Fourier transform, allows the frequency components of a signal to be revealed
Fourier series approximation Frequency Component AnotherThese signals can be discussed with constant representation respect to the time domainof the orFourier the series 1 Hz + frequency domain. approximationA sine (or cosine) waveform is a = single frequency; 2 Hz + any other waveform can be the sum of a number of sine and 3 Hz + cosine waves.
Fourier series approximation Fourier Transform
y(t) = A + B1sin(f1t) + C1cos(f1t) + B2sin(f2t) + C2cos(f2t) + … • Converts signals from the time domain to the number of harmonics frequency domain used Approximation of the data improves • The Fourier transform separates a signal into as the number of sinusoids of different frequency which sum to the harmonics used original waveform: it distinguishes the different increases frequency sinusoids and their respective amplitudes However, 94% of – Discrete Fourier Transform (DFT) the total signal – Fast Fourier Transform (FFT) power was contained in just • Typically, the square of the FFT is used instead, the 1st harmonic which is called the Power Density Spectrum
3 Fourier Transform Fourier Transform original signal 20 Hz noise added Raw EMG signal Power Spectral Density
time (s) time (s)
FFT FFT
samples frequency (Hz)
frequency (Hz) frequency (Hz)
Frequency Content ADC – Temporal Resolution
Frequency content of some common signals Now, back to sampling rate: encountered in biomechanics: • What happens if you sample at a rate lower than the Nyquist frequency? Activity Max Freq of Interest • An error, known as aliasing, will occur, corrupting
Standing posture 3 Hz your data (aliased signals show up at a freq of fsamp – fsig, so 40Hz signal sampled at 50HZ will show up as a 10Hz signal) Walking 6 Hz Running 10-15 Hz • When aliasing occurs, frequencies greater than the sampling rate “fold back” into the lower Heelstrike transient 100-300 Hz frequency components, distorting the signal EMG (surface) 400-500 Hz • High frequency information is not simply lost, it EMG (indwelling) 800-1000 Hz actually reappears as false low frequency content
Sampling Sampling
• “One common form of aliasing, which most people have observed, is the 'wagon wheels' effect in films. Since the rate of the film is usually much lower than the frequency at which the spokes of a wheel pass any one point, aliasing takes place and the wheels appear to turn either much more slowly than they really are, or even seem to go backwards sometimes." (D.W. Grieve et. al., Techniques for the Analysis of Human Movement. Princeton Book Company, 1975, p. 41.)
4 4 Hz sine wave Sampling Sampling analog Frequency
<2hz 50 This is a 4 Hz Hz signal, so the Nyquist freq is 25 8 Hz Hz
10hz 12.5 Hz
Aliasing occurs at 6.25 sampling rate Hz of 6.25 Hz
Sampling Frequency Sampling Rate Take home message(s):
• Sampling at the Nyquist frequency ensures that frequency content of signal is preserved
• In practice, sampling at 4-5 times the highest frequency present in the signal ensures that signal amplitude characteristics are preserved as well
• When possible, an “anti-aliasing” filter should be used to eliminate noise that exists above the sampling rate Another example of aliasing: 1 Hz signal appears to be 0.33 Hz
Sampling Rate Sampling
Typical sampling rates used to collect various • How do we determine the length of signals in biomechanics: sampling period? Signal Typical SR • In repetitive motion the time needs to be more than the time for 1 cycle so if gait is Motion 30-240 Hz 120 steps (60 strides) per minute then the Force 100-1000 Hz rate is 1Hz and you need to sample longer EMG (surface) 1000-2000 Hz than 1s to obtain the full data set. EMG (indwelling) 2000-5000 Hz • You may increase time to collect multiple repetitions.
5 Sampling Measurement Errors
• In non repetitive activities such as • No measurement process is perfect, there will be some degree of error in the data lifting then you need to go from start to finish. • Measurement errors fall into two categories – Random Errors • In cases where frequency of movement – Systematic Errors is not obvious pilot data needed to enable appropriate decision. Winter • Both can be minimized, but neither can be completely eliminated reported that in quiet standing significant data found at .002Hz which • In particular, random errors may be reduced through data processing techniques, systematic would require an 8 minute trial. errors generally can not
Random Error - Example Systematic Error - Example 3 people measure the distance between two lines 1 person measures the distance between two lines that that are 1.5 cm apart are 1.5 cm apart, but places the ruler down incorrectly
0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2
1.5 cm 1.4 cm 1.6 cm 1.2 cm 1.2 cm 1.2 cm
They don’t all get the same value, but this “random This individual was very consistent in his fluctuation” can be largely eliminated by averaging measurements, but he has introduced a the three values systematic error into the data (1.5 + 1.4 + 1.6) / 3 = 1.5 cm No amount of processing can remove this error
Measurement Errors in Minimizing Measurement Biomechanics Errors
Systematic Errors: cover frequency spectrum • Use of consistent, proper techniques for – Poor placement of anatomical markers data collection – Movement of markers/skin relative to bones – Calibration nonlinearities • Use of 3-D techniques for motion capture – Digitization errors (some) • Proper training of personnel Random Errors: typically high frequency – Digitization errors (some) • Use of high quality equipment – Marker jiggling/vibration – Electrical interference
6 Up Next…
Topic – Kinematics
Readings – Robertson et al. (2004) Chapter 1, pp. 9 – 34 – Winter (1990) Chapter 2, pp. 11 – 48
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